By Klee V. (ed.)
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Extra resources for Convexity
The expected value of X, denoted E[X], is a summary measure indicating the center of the distribution FX, and it is sometimes called the mean of X. The “E” is called the expectation operator. If X is a discrete-valued random variable taking values a1 , a2 ,…. (possibly infinite) with mass function pX, then a weighted average of the possible values of X with each value weighted by its probability of being generated. For example, if X has a Bernoulli distribution with parameter 0 < γ < 1, then its mass function is and the E[X] = 0 · pX(0) + 1 · pX(1) = γ If Y is a continuous-valued random variable with density fY, then the integral of the possible values of Y with each value weighted by the density fY.
A service-system example is used to illustrate the approach, but sample-path decomposition can characterize the behavior of many types of systems, including those described in Chapter 1. Sample- path decomposition is also a convenient way to formulate mathematical models of systems, models that can be used to evaluate how changes will affect existing systems or predict the performance of systems that do not yet exist. Formulation and analysis of these models is the topic of this book, and sample-path decomposition is the perspective that we employ throughout.
Perhaps these 20 customers are representative of the customers that The Darker Image serves every day, but 20 customers do not even constitute a full day’s work. And the particular pattern of inputs observed (times and types of service) will likely never be repeated, and may even differ from store to store. The fact that full-service customers have longer service times must sometimes imply an advantage for self-service customers if they have a dedicated copier. The problem is that we generated just one possible sample path for this system.